So I'm trying to create a homebrew feat. An option for the players should be to gain proficiency in any 1 musical instrument of their choice. But there are barely 10 options available while a quick search in Game Rules -> Equipment shows a much larger number of instruments.

I'm using the modifier -> Choose a Musical Instrument. I don't see any that would let a player pick something like, e.g., wargong while limiting to instruments at the same time.

There are 10 instruments in the PHB which are mostly generic "types".

The additional instruments you detail were introduced in SCAG and are more specific variations of these types. For example, "birdpipes" are a type of pan pipe, so you'd just pick pan pipes because they're played the same way, while Glaur is a type of horn so you'd just pick horn. Tambourine is a type of drum, as is a wargong, so just pick drum since knowing how to play one will pretty much let you know how to play the other - the sound may be different but understanding how to play it is the same. A whistlecane is a type of flute, anyone who is proficient with flutes will be proficient with whistlecanes.. And so on.

The PHB listings, which is what you pick in feats etc, are your skills with those types of instruments. The instruments presented in SCAG are more specific flavouring. You can add the SCAG instruments to characters as equipment. 

It would be unrealistic to include specific proficiencies for specific instruments since it doesn't work that way in real life - skills are transferable for like-instruments. Knowing how to play a viol is the same as knowing how to play violin - the sound is just a bit different. Knowing how to play a dulcimer is the same as playing a recorder (because a modern recorder is a variation of dulcimer), etc. There are many instruments that are just variations of each other to produce different sound while being played in basically the same way. 

In your example of wargong - this is a type of drum. With drums it's about knowing how hard and how fast to hit the drum to produce the sound you want from it, and this applies to any kind of drum be it traditional drums, tambourines, gongs, and whatnot. 

All the SCAG specific instruments are covered by the PHB instrument proficiency types, so just pick the right proficiency type for the instrument you want to play. 

Yes, the Basic Rules is a wholly different source than the PHB. The Basic Rules and all of its content is free, the PHB is not. That’s why only 1 subclass for each class is available for free, why only one subrace for each race is free, why only one feat is free, and why only about half the spells from he PHB are free. That free content is what appears in the Basic Rules, the rest only appears in the PHB.

This really isn't a bug. They can't list every possible instrument that ever existed. Each source has only a selection.

The basic list is very generalized. (Wargong can be found in SCAG IIRC. I don't have SCAG, though.)

Basic Rules has "viol" which, historically, was played two different ways, but the modern reference to "viol" most often means upright (da gamba) like a violoncello (or just cello) instead of like a viola (or violin) on the shoulder or under the chin (da braccio). Yet, by calling it "viol", people can technically choose how they want the character to look when playing it—like a violin or like a cello. Viol can describe a whole family of instruments (though carrying something the size of a cello is asking for a replacement cost every time it's taken on an adventure).

Pick something in the realm of the instrument you're seeking and add its description elsewhere in the character sheet. (There are many places such descriptions could go.) If you want a wargong and it's not listed in the content that you have, you can use "drum" and describe it elsewhere. (Drums can represent any unpitched or pitched percussion instrument. Even the modern side drum can be pitched while playing if you know how to do it though it's mostly used unpitched today.)